Question

In the population, the mean SAT score is 1000 . Would you be more likely (or equally likely) to get a sample mean of 1200 if you randomly sampled 10 students or if you randomly sampled 30 students? Explain.

Short answer

You are more likely to get a sample mean of 1200 by sampling 10 students.

Step-by-step solution

Understand the Concept of Sampling Distribution

The challenge here is to comprehend how sample size impacts the distribution of the sample mean. As sample size increases, the sampling distribution of the sample mean becomes more concentrated around the population mean, according to the Central Limit Theorem.

Define the Standard Error Formula

The standard error (SE) is calculated as the population standard deviation (σ) divided by the square root of the sample size (n). The formula is \[SE = \frac{\sigma}{\sqrt{n}}\]where \( \sigma \) is the standard deviation of the population.

Discuss the Influence of Sample Size

For a small sample size (e.g., 10 students), the standard error is larger, making the sample mean more variable and often requiring a wider range to capture. For a larger sample size (e.g., 30 students), the standard error is smaller, resulting in a tighter distribution around the mean.

Analyze Sample Means

A sample mean of 1200 is considerably higher than the population mean of 1000. As the sample size increases, the probability of observing a sample mean far from the population mean, such as 1200, decreases due to the smaller standard error.

Conclude Which Sample Size is More Likely to Give a Mean of 1200

Considering the above points, you are more likely to obtain a sample mean of 1200 by sampling 10 students than by sampling 30 students, because the variability in smaller samples makes extreme values more probable.

Key concepts

Sampling distribution

Sampling distribution refers to the probability distribution of a given statistic based on a random sample. When we talk about the sample mean, the sampling distribution helps us understand the spread and behavior of the sample means when drawn from the same population repeatedly.

The Central Limit Theorem (CLT) plays a crucial role here. It states that, regardless of the population distribution, the sampling distribution of the sample mean will approach a normal distribution as the sample size becomes larger.

Let's say we repeatedly draw samples of students' SAT scores. Each sample might have slightly different average scores, resulting in different sample means. These means form the sampling distribution.

• For smaller samples, the distribution tends to be wider and less symmetric because they capture a higher variability from the population mean.
• For larger samples, the sampling distribution tends to narrow, centering more closely around the actual population mean of 1000 in our example.

Standard error

The standard error (SE) is a statistical term that quantifies the amount of variability or dispersion of a set of sample means from the population mean. It's a reflection of how much the sample mean would differ from the population mean if you were to take multiple samples.

Using the formula \[SE = \frac{\sigma}{\sqrt{n}}\]where \(\sigma\) is the population standard deviation and \(n\) is the sample size, you can see that the standard error decreases as the sample size increases.

This means:

• Larger samples give a smaller SE, showing less variability and indicating that the sample mean is likely closer to the population mean.
• Smaller samples lead to a larger SE, demonstrating more variability and a wider spread of sample means.

In simple terms, the standard error helps us understand how precise our estimate of the population mean is. A smaller standard error implies a more reliable and accurate estimate.

Sample size effect

The sample size effect is the impact that the number of observations in a sample has on the characteristics of the sample, such as its variability and reliability.

As the sample size increases, two key effects occur:

• The standard error decreases, which means that sample means are closer to the population mean.
• The sampling distribution of the mean becomes narrower and more symmetric.

The impact of this is significant:

• With a sample size of 10, we're likely to observe more variability, and extreme sample means like 1200 are more probable due to a larger standard error.
• With a sample size of 30, the smaller standard error leads to less variability and a lower likelihood of seeing a sample mean far from the population mean, such as 1200.

Thus, larger samples are generally more reliable and provide a better approximation of the population characteristics, but they reduce the chance of capturing extreme values in the sample mean.